ME338A CONTINUUM MECHANICS lecture notes 15 tuesday, march 2nd, 2010 5 Introduction to Nonlinear Continuum Mechanics 5.1 Kinematics 5.1.1 Motion and Configurations Consider a material body B composed of infinitely many ma- terial points P B that are identified with geometrical points ∈ in the three-dimensional Euclidean space 3. R 3 χt(P) R P xt e2 e1 t B B e3 The configuration of the body B in 3 at time t is described R by the one-to-one relation 3 B t , χt : → B ∈ R (5.1.1) P B x = χ (P) . ∈ 7→ t t ∈ Bt Having this definition at hand, we go on to describe the mo- tion of the body as a family of configurations parameterized 125 5 Introduction to Nonlinear Continuum Mechanics P B χt0 (P) χt4 (P) 3 χt1 (P) R χt3 (P) χt2 (P) Path by time in the Euclidean space 3. Throughout the motion, R the material point P occupies a sequence of places in space, which forms the path (trajectory) of the material point. P B χt0 (P) χt(P) 3 1 R ϕt(X) := χt χ− (X) ◦ t0 v(x, t) X x ϕX (t) B S While describing the motion of a solid body, it is common practice to name its placement at time t0 as the reference con- figuration that generally possesses an undistorted stress-free state and is henceforth denoted as χ (B). Likewise, the B ≡ t0 configuration of the body at current time t is hereafter de- noted as χ (B). The reference and the spatial positions S ≡ t occupied by a material point P within the Euclidean space 126 5 Introduction to Nonlinear Continuum Mechanics are labeled by the reference coordinates X := χ (P) and t0 ∈ B the spatial coordinates x := χ (P) , respectively. In order t ∈ S to describe the motion of the body in the Euclidean space, we introduce a non-linear deformation map ϕt(X) between χt0(P) and χt(P) , ϕt(X) : B → S (5.1.2) 1 X x = ϕ (X) := χ χ− (X) 7→ t t ◦ 0 that maps the material points X onto their deformed ∈ B spatial positions x = ϕ (X) at time t . Since t ∈ S ∈ R+ the deformation map ϕt(X) is bijective, it can be inverted uniquelly to obtain the inverse deformation map 1 , ϕ− (x) : S → B (5.1.3) t 1 1 x X = ϕ− (x) := χ χ− (x) . 7→ t 0 ◦ t In indicial notation, we have 1 xa = ϕa(X1, X2, X3, t) and XA = ϕ−A (x1, x2, x3, t) . (5.1.4) Although we use a common coordinate system to describe the motion of the body, it will be convenient to assign the lower-case indices to tensorial fields belonging to the cur- rent configuration and the upper-case indices for referential fields. 127 5 Introduction to Nonlinear Continuum Mechanics Examples: Simple extension of cube e2 1 1 x1 = λ1(t)X1 λ2 e1 1 x2 = λ2(t)X2 λ3 λ1 x3 = λ3(t)X3 e3 Simple shear e2 γ x1 = X1 + γ(t)X2 1 x2 = X2 e1 x3 = X3 1 Pure shear 1 1 e2 λ+λ− λ λ− x1 = 2 X1 + −2 X2 1 1 λ+λ 1 λ λ− λ+λ− − 1 x2 = −2 X1 + 2 X2 2 λ λ 1 − − e 2 1 x = X 1 3 3 λ Periodic deformation e2 α x1 = X1 + α(t) sin(2πX2) + a 1 x2 = X2 α(t) sin(2πX1) e1 − x3 = X3 1 a α 128 5 Introduction to Nonlinear Continuum Mechanics 5.1.2 Velocity and Acceleration Recall the definition of motion in (5.1.2), x := ϕ(X, t) = ϕt(X) = ϕX (t) (5.1.5) where ϕ (X) denotes the configuration of at time t and t B ϕX (t) is the path of the particle P. 5.1.2.1 Material Velocity and Acceleration We can now define the material velocity d V (X) := ∂ ϕ(X, t) = ϕ (t) , (5.1.6) t t dt X and the material acceleration d A (X) := ∂ V(X, t) = V (t) (5.1.7) t t dt X of the motion. It is important to notice that both the material velocity V(X, t) and the material acceleration A(X, t) belong to the current configuration but are functions of material S coordinates X and time t. 5.1.2.2 Spatial Velocity and Acceleration The spatial counterparts of the rates of material description of the motion can be obtained by reparameterizing the func- tional dependency of these fields in terms of spatial coor- dinates x. The spatial velocity and the spatial acceleration are then defined as 1 1 v(x, t) := V(ϕ− (x), t) = V (X) ϕ− (x) , (5.1.8) t t ◦ t and 1 1 a(x, t) := A(ϕ− (x), t) = A (X) ϕ− (x) , (5.1.9) t t ◦ t respectively. The path ϕX (t) is called the integral curve of v. 129 5 Introduction to Nonlinear Continuum Mechanics 5.1.2.3 Material Time Derivative of Spatial Fields Consider a spatial field f (x, t) : ϕ ( ) . t B × R+ → R The material time derivative of f (x, t) is defined as D ∂ f (x, t) = f˙(x, t) := f (x, t) + f v (5.1.10) Dt ∂t ∇x · For F(X, t) := f (x, t) ϕ (X), the following equality ◦ t ∂ F(X, t) = f˙(x, t) (5.1.11) ∂t holds. An immediate example is the spatial acceleration a(x, t) that is none other than the material time derivative of the spatial velocity 1 a(x, t) := A (X) ϕ− (x) = ∂ v + v v . (5.1.12) t ◦ t t ∇x · where v stands for the spatial gradient of velocity. ∇x 130 5 Introduction to Nonlinear Continuum Mechanics Example: Rigid Body Motion Chadwick (1976), p.52 The rigid body motion of a body is described by x = ϕ(X, t) = c(t) + Q(t) X , · where Q is proper orthogonal, i.e. Q QT = 1 and detQ = · +1. In rigid motion the distance between two points does not change. That is, for T := X X and t := x x = B − A B − A Q T with x A,B = ϕ(X A,B , t), we end up with the identity · { } { } t := √t t = (Q T) (Q T) = T (QT Q T) = T . || || · q · · · q · · · || || The motion can be then inverted to obtain 1 T X = ϕ− (x) = Q (x c) . t · − The material velocity and material acceleration V (X) := ∂ ϕ(X, t) = c˙ + Q˙ X t t · A (X) := ∂ V(X, t) = c¨ + Q¨ X t t · The spatial velocity and spatial acceleration 1 T v(x, t) := V(ϕ− (x), t) = c˙ + Q˙ Q (x c) t · · − 1 T a(x, t) := A(ϕ− (x), t) = c¨ + Q¨ Q (x c) t · · − Since Q QT = 1, we have · T Q˙ QT = Q Q˙ = (Q˙ QT)T · − · − · Therefore, Ω := Q˙ QT = ΩT · − 131 5 Introduction to Nonlinear Continuum Mechanics is a skew-symmetric tensor, which has an associated axial vector ω 3 defined by ∈ R Ω β = ω β β 3 . · × ∀ ∈ R Furthermore, it can be shown that T Q¨ QT = Ω˙ Q˙ Q˙ = Ω˙ + Ω2 , · − · Q¨ QT β = Ω˙ β + Ω2 β = ω˙ β + ω (ω β) β 3 . · · · · × × × ∀ ∈ R Incorparating these results in the spatial velocity and spatial acceleration expressions for β (x c), we end up with ≡ − v(x, t) = c˙ + ω (x c) × − a(x, t) = c¨ + ω˙ (x c) + ω [ω (x c)] × − × × − The latter can also be obtained by taking the material time derivative of the spatial velocity field as described in (5.1.12). a(x, t) = ∂ v + v v t ∇x · = ∂ [c˙ + ω (x c)] t × − + [c˙ + ω (x c)] [c˙ + ω (x c)] ∇x × − · × − = c¨ + ω˙ (x c) ω c˙ × − − × + ω [c˙ + ω (x c)] × × − = c¨ + ω˙ (x c) + ω [ω (x c)] × − × × − 132 5 Introduction to Nonlinear Continuum Mechanics 5.1.3 Fundamental Geometric Maps This section is devoted to the geometric mapping of basic tangential, normal and volumetric objects. 5.1.3.1 Deformation Gradient. Tangent Map ϕt(X) t T X F := ϕ (X) ∇X t x TX Tx B Θ S B χ0( ) χt(Θ) S Θ R Probably the most fundamental deformation measure used in kinematics of finite deformation is the deformation gradi- ent. It can be considered as a linear map of the referential tangent vectors onto the spatial counterparts. To this end, let χ0(Θ) and χt(Θ) be the material and spatial curves param- eterized by a common variable Θ on and , respec- ∈ R B S tively. It is important to observe that the spatial curve is related to the reference curve by the non-linear deformation map χt(Θ) = ϕt(χ0(Θ)). Tangents of the curves belonging to the respective tangent spaces are defined as dχ (Θ) dχ (Θ) T := 0 T and t := t T (5.1.13) dΘ ∈ XB dΘ ∈ x S Through the chain rule, we obtain dχ (Θ) dχ (Θ) t = t = ϕ (X) 0 = F T, (5.1.14) dΘ ∇X t · dΘ · 133 5 Introduction to Nonlinear Continuum Mechanics where we introduced the deformation gradient F := ϕ (X) ∇X t as the Frechet´ derivative of the deformation. t T X F := ϕ (X) ∇X t x T T XB x S The deformation gradient is also called the tangent map T T , F(X, t) : XB → x S (5.1.15) T t = F T , 7→ · between the tangent spaces T and T of the configura- XB x S tions (manifolds) and , respectively.